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Expansion of (1 - x^3) * (1 - x^5) * (1 - x^6) / (1 - x^15) in powers of x.
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%I #20 Sep 08 2022 08:45:56

%S 1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,

%T 1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,

%U 1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1

%N Expansion of (1 - x^3) * (1 - x^5) * (1 - x^6) / (1 - x^15) in powers of x.

%H G. C. Greubel, <a href="/A188578/b188578.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1,-1,0,1,-1).

%F Euler transform of length 15 sequence [0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1].

%F a(n) = b(2*n + 1) where b(n) is multiplicative with b(p^e) = 0^e if p<7, b(p^e) = 1, if p == 1, 17, 19, 23 (mod 30), b(p^e) = (-1)^e if p == 7, 11, 13, 29 (mod 30).

%F G.f.: (1 - x^3) * (1 - x^5) * (1 - x^6) / (1 - x^15).

%F a(-1 - n) = -a(n).

%F G.f.: (1-x)^2 *(1+x) *(1+x+x^2) *(1-x+x^2) / (1-x+x^3-x^4+x^5-x^7+x^8). - _R. J. Mathar_, Apr 09 2011

%F a(n) = -a(-1-n) = a(n+15) for all n in Z. - _Michael Somos_, May 21 2015

%F a(2*n) = a(n-4), a(2*n + 1) = a(n+4), a(3*n) = A080891(n+1), a(3*n + 1) = 0, a(3*n + 2) = -A080891(n) for all n in Z. - _Michael Somos_, May 21 2015

%e G.f. = 1 - x^3 - x^5 - x^6 + x^8 + x^9 + x^11 - x^14 + x^15 - x^18 - x^20 + ...

%e G.f. = q - q^7 - q^11 - q^13 + q^17 + q^19 + q^23 - q^29 + q^31 - q^37 - q^41 + ...

%t a[ n_] := KroneckerSymbol[ -60, 2 n + 1];

%o (PARI) {a(n) = kronecker( -60, 2*n + 1)};

%o (Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x^3)*(1-x^5)*(1-x^6)/(1-x^15))); // _G. C. Greubel_, Aug 13 2018

%Y Cf. A080891.

%K sign,easy

%O 0,1

%A _Michael Somos_, Apr 04 2011